On the Fibonacci numbers and their sums which are close to a power of 3

Abstract

Let (Ln)n ≥ 0 be the Lucas sequence defined by Ln+2 = Ln+1 + Ln for all n ≥ 0, with initial values L0 = 2 and L1 = 1. In this paper, we find all the Fibonacci numbers 2Fn and sums of two Fibonacci numbers which are close to a power of 3. As a corollary, we determine all Lucas numbers close to a power of 3. To prove our results, we will use Baker's theory lower bound for linear forms in logarithms of algebraic numbers, properties of continued fractions, and a version of the Baker-Davenport reduction method in Diophantine approximation.